The proximal point algorithm (PPA) is classical, and it is implicit in the sense that the resulting proximal subproblems may be as difficult as the original problem. In this paper, we show that with appropriate choices of proximal parameters, the application of PPA to the linearly constrained convex programming can result in easy proximal subproblems. In particular, under some practical assumptions on the objective function, these proximal subproblems become explicit in the sense that they all have closed-form solutions or can be efficiently solved up to a high precision. We thus present some implementable contraction methods with explicit proximal regularization for linearly constrained convex programming, and their global convergence is proved easily under the analytic framework of contraction type methods.